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| In [[mathematics]], the '''Kolmogorov continuity theorem''' is a [[theorem]] that guarantees that a [[stochastic process]] that satisfies certain constraints on the [[moment (mathematics)|moments]] of its increments will be continuous (or, more precisely, have a "continuous version"). It is credited to the [[Soviet Union|Soviet]] [[mathematician]] [[Andrey Kolmogorov|Andrey Nikolaevich Kolmogorov]].
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| ==Statement of the theorem==
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| Let <math>X : [0, + \infty) \times \Omega \to \mathbb{R}^{n}</math> be a stochastic process, and suppose that for all times <math>T > 0</math>, there exist positive constants <math>\alpha, \beta, K</math> such that
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| :<math>\mathbb{E} \left[ | X_{t} - X_{s} |^{\alpha} \right] \leq K | t - s |^{1 + \beta}</math>
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| for all <math>0 \leq s, t \leq T</math>. Then there exists a continuous version of <math>X</math>, i.e. a process <math>\tilde{X} : [0, + \infty) \times \Omega \to \mathbb{R}^{n}</math> such that | |
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| * <math>\tilde{X}</math> is [[sample continuous process|sample continuous]];
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| * for every time <math>t \geq 0</math>, <math>\mathbb{P} (X_{t} = \tilde{X}_{t}) = 1.</math>
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| ==Example==
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| In the case of [[Brownian motion]] on <math>\mathbb{R}^{n}</math>, the choice of constants <math>\alpha = 4</math>, <math>\beta = 1</math>, <math>K = n (n + 2)</math> will work in the Kolmogorov continuity theorem.
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| ==References==
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| * {{cite book | author=Øksendal, Bernt K. | authorlink=Bernt Øksendal | title=Stochastic Differential Equations: An Introduction with Applications | publisher=Springer, Berlin | year=2003 | isbn=3-540-04758-1}} Theorem 2.2.3
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| [[Category:Stochastic processes]]
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| [[Category:Statistical theorems]]
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| [[Category:Probability theorems]]
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Andrew Simcox is the title his parents gave him and he totally loves this title. Office supervising is exactly where her main income arrives from but she's already utilized for another 1. Her family lives in Ohio. As a woman what she truly likes is style and she's been doing it for quite a whilst.
Here is my page ... psychics online